1 00:00:07,280 --> 00:00:12,710 PROFESSOR: OK, this lecture, this day, is differential 2 00:00:12,710 --> 00:00:15,350 equations day. 3 00:00:15,350 --> 00:00:22,380 I just feel even though these are not on the BC exams, that 4 00:00:22,380 --> 00:00:25,140 we've got everything we need to actually 5 00:00:25,140 --> 00:00:27,670 see calculus in use. 6 00:00:27,670 --> 00:00:31,910 We've got the derivatives of the key functions and ready 7 00:00:31,910 --> 00:00:34,050 for a differential equation. 8 00:00:34,050 --> 00:00:36,710 And there it is. 9 00:00:36,710 --> 00:00:38,460 When I look at that equation-- 10 00:00:38,460 --> 00:00:41,830 so it's a differential equation because it has the 11 00:00:41,830 --> 00:00:46,700 derivatives of y as well as y itself in the equation. 12 00:00:46,700 --> 00:00:52,740 And when I look at it, I see it's a second order equation 13 00:00:52,740 --> 00:00:55,610 because there's a second derivative. 14 00:00:55,610 --> 00:01:03,140 It's a linear equation because second derivative, first 15 00:01:03,140 --> 00:01:07,660 derivative, and y itself are separate, no multiplying of y 16 00:01:07,660 --> 00:01:09,490 times y prime. 17 00:01:09,490 --> 00:01:12,030 In fact, the only multiplications are by these 18 00:01:12,030 --> 00:01:19,440 numbers m, r and k, and those are constant numbers coming 19 00:01:19,440 --> 00:01:21,610 from the application. 20 00:01:21,610 --> 00:01:25,120 So I have a constant coefficient, linear, second 21 00:01:25,120 --> 00:01:29,680 order, differential equation, and I'd like to solve it. 22 00:01:29,680 --> 00:01:34,080 And we can do it because it uses the very functions that 23 00:01:34,080 --> 00:01:36,880 we know how to find derivatives of. 24 00:01:36,880 --> 00:01:40,580 Let me take two or three special cases where those 25 00:01:40,580 --> 00:01:43,550 functions appear purely. 26 00:01:43,550 --> 00:01:46,640 So one special case is if I knock out the second 27 00:01:46,640 --> 00:01:52,450 derivative term, and let me just choose-- 28 00:01:52,450 --> 00:01:59,410 rewrite it in the way it looks easiest and best. It's just 29 00:01:59,410 --> 00:02:02,260 the derivative of y as some multiple of y. 30 00:02:02,260 --> 00:02:07,000 It's now first order, and we know the function that solves 31 00:02:07,000 --> 00:02:08,889 that equation. 32 00:02:08,889 --> 00:02:12,130 When the derivative equals the function itself, that's the 33 00:02:12,130 --> 00:02:13,640 exponential. 34 00:02:13,640 --> 00:02:17,120 If I want to have an extra factor a here, then I 35 00:02:17,120 --> 00:02:21,680 need e to the at. 36 00:02:25,900 --> 00:02:31,270 And usually, in fact, we expect that with a first order 37 00:02:31,270 --> 00:02:34,860 equation, in the solution there'll be some constant that 38 00:02:34,860 --> 00:02:38,770 we can set later to match the starting condition. 39 00:02:38,770 --> 00:02:41,340 And that constant shows up here. 40 00:02:41,340 --> 00:02:45,730 It's just if e to the at solves that equation, as it 41 00:02:45,730 --> 00:02:49,830 does, because when I take the derivative, down comes on a, 42 00:02:49,830 --> 00:02:52,520 so does c times e to the at. 43 00:02:52,520 --> 00:02:56,790 That's because the equation is linear. 44 00:02:56,790 --> 00:02:58,520 So that's a nice one. 45 00:02:58,520 --> 00:03:00,720 Pure exponential. 46 00:03:00,720 --> 00:03:03,020 OK, ready for this one. 47 00:03:03,020 --> 00:03:07,700 So this one, I don't have this middle term. 48 00:03:07,700 --> 00:03:10,860 I just have the second derivative and the function. 49 00:03:10,860 --> 00:03:17,400 And let me again change to letters I like, putting the ky 50 00:03:17,400 --> 00:03:21,490 on the other side with a minus sign. 51 00:03:21,490 --> 00:03:33,290 So this omega squared will be k/m when I 52 00:03:33,290 --> 00:03:35,260 reorganize that equation. 53 00:03:35,260 --> 00:03:40,580 My point is we can solve this equation, the second 54 00:03:40,580 --> 00:03:45,080 derivative equaling minus the function. 55 00:03:45,080 --> 00:03:46,740 We've met that. 56 00:03:46,740 --> 00:03:49,600 That's this equation that the sine and 57 00:03:49,600 --> 00:03:52,640 the cosine both solve. 58 00:03:52,640 --> 00:03:57,320 There we get two solutions for this second order equation. 59 00:03:57,320 --> 00:04:01,170 And just as with the a in this problem, so with this number 60 00:04:01,170 --> 00:04:04,780 here, I'll just jiggle the sine and the cosine a little 61 00:04:04,780 --> 00:04:11,570 bit so that we get omega to come down twice when I take 62 00:04:11,570 --> 00:04:13,040 two derivatives. 63 00:04:13,040 --> 00:04:17,269 So the solution here will be-- 64 00:04:17,269 --> 00:04:23,570 one solution will be the cosine of omega t because two 65 00:04:23,570 --> 00:04:26,670 derivatives of the cosine is minus the cosine-- 66 00:04:26,670 --> 00:04:28,240 that's what this asks for-- 67 00:04:28,240 --> 00:04:31,320 with the factor omega coming out twice. 68 00:04:31,320 --> 00:04:37,400 And another solution will be sine of omega t 69 00:04:37,400 --> 00:04:39,040 for the same reason. 70 00:04:39,040 --> 00:04:43,250 And again, now with a second order equation, I'm expecting 71 00:04:43,250 --> 00:04:46,520 a couple of constants to be able to choose later. 72 00:04:46,520 --> 00:04:53,170 And here they are: c cosine omega t and d sine omega t. 73 00:04:53,170 --> 00:04:57,300 That's the general solution to that equation. 74 00:04:57,300 --> 00:04:59,120 So we know that. 75 00:04:59,120 --> 00:05:01,820 And these are the two important ones. 76 00:05:01,820 --> 00:05:03,870 There's another less important one and an 77 00:05:03,870 --> 00:05:05,560 extremely simple one. 78 00:05:05,560 --> 00:05:10,530 Suppose all that went away, and I just had as a third very 79 00:05:10,530 --> 00:05:14,695 special case d second y dt squared equals 0. 80 00:05:17,850 --> 00:05:20,930 We sure know the solution to that. 81 00:05:20,930 --> 00:05:25,220 What functions have second derivative equals 0? 82 00:05:25,220 --> 00:05:27,300 Well, a constant function does, certainly. 83 00:05:29,810 --> 00:05:32,610 A constant, even its first derivative is 0 84 00:05:32,610 --> 00:05:34,130 much less its second. 85 00:05:34,130 --> 00:05:39,690 And then t does. 86 00:05:39,690 --> 00:05:42,520 Its first derivative is 1 and then the second 87 00:05:42,520 --> 00:05:43,760 derivative is 0. 88 00:05:43,760 --> 00:05:48,660 So there show up the powers of t. 89 00:05:48,660 --> 00:05:52,770 Well, the first two powers, t to the 0 and t to the 1, show 90 00:05:52,770 --> 00:05:56,120 up in that very special case. 91 00:05:56,120 --> 00:05:58,360 Sine and cosine show up here. 92 00:05:58,360 --> 00:06:00,200 e to the at shows up here. 93 00:06:00,200 --> 00:06:04,300 And now let me tell you the good part of this lecture. 94 00:06:04,300 --> 00:06:09,190 The solution to this equation and in fact to equations of 95 00:06:09,190 --> 00:06:15,790 third, fourth, all orders, are products of these ones that we 96 00:06:15,790 --> 00:06:22,900 know: exponentials times sines and cosines times powers of t. 97 00:06:22,900 --> 00:06:26,020 That's all we need to solve constant coefficient 98 00:06:26,020 --> 00:06:27,580 differential equations. 99 00:06:27,580 --> 00:06:31,290 So I plan now to go ahead and solve the-- 100 00:06:31,290 --> 00:06:36,580 and move toward this equation. 101 00:06:36,580 --> 00:06:40,950 I should have said that's a fundamental equation of 102 00:06:40,950 --> 00:06:42,760 engineering. 103 00:06:42,760 --> 00:06:44,990 m stands for some mass. 104 00:06:44,990 --> 00:06:48,090 Oh yeah, let me draw a picture, and you'll see why I 105 00:06:48,090 --> 00:06:54,070 chose to choose t rather than x, because things are 106 00:06:54,070 --> 00:06:55,670 happening in time. 107 00:06:55,670 --> 00:06:57,270 And what is happening? 108 00:06:57,270 --> 00:06:58,570 Let me show you. 109 00:06:58,570 --> 00:07:01,990 What's happening in time is typically this would be a 110 00:07:01,990 --> 00:07:09,380 problem with us some kind of a spring hanging down, and on 111 00:07:09,380 --> 00:07:11,870 that spring is a mass m. 112 00:07:15,250 --> 00:07:20,740 OK, so what happens if I-- 113 00:07:20,740 --> 00:07:24,650 I've pulled that mass down, so I've stretched the spring, and 114 00:07:24,650 --> 00:07:27,020 then I let go. 115 00:07:27,020 --> 00:07:29,410 Then what does the spring do? 116 00:07:29,410 --> 00:07:34,400 Well, the spring will pull the mass back upwards, and it will 117 00:07:34,400 --> 00:07:39,920 pull it back up to the point where it squeezes, compresses 118 00:07:39,920 --> 00:07:40,590 the spring. 119 00:07:40,590 --> 00:07:42,420 The spring will be-- 120 00:07:42,420 --> 00:07:43,980 like instead of being stretched out, 121 00:07:43,980 --> 00:07:45,350 it'll be the opposite. 122 00:07:45,350 --> 00:07:47,140 It'll be compressed in. 123 00:07:47,140 --> 00:07:51,430 And then when compressed in, it'll push the mass. 124 00:07:51,430 --> 00:07:53,820 Being compressed, the spring will push. 125 00:07:53,820 --> 00:07:59,140 It'll push the mass down again and up again, and I get 126 00:07:59,140 --> 00:08:01,030 oscillation. 127 00:08:01,030 --> 00:08:04,130 Oscillation is what I'm seeing here. 128 00:08:04,130 --> 00:08:09,963 And what are examples of oscillation in real life? 129 00:08:09,963 --> 00:08:20,640 The spring or a clock, especially a grandfather clock 130 00:08:20,640 --> 00:08:24,060 that's swinging back and forth, back and forth, so I'll 131 00:08:24,060 --> 00:08:25,310 just put a clock. 132 00:08:29,290 --> 00:08:32,710 Music, a violin string is oscillating. 133 00:08:32,710 --> 00:08:36,015 That's where the beautiful sound comes from. 134 00:08:39,626 --> 00:08:44,390 Our heart is in and out, in and out, a regular 135 00:08:44,390 --> 00:08:45,640 oscillation. 136 00:08:50,160 --> 00:08:52,310 I could add molecules. 137 00:08:52,310 --> 00:08:55,520 They oscillate extremely quickly. 138 00:08:55,520 --> 00:09:02,780 So this equation that I'm aiming for comes up in 139 00:09:02,780 --> 00:09:05,310 biology, in chemistry-- 140 00:09:05,310 --> 00:09:08,490 for molecules, it's a very important equation-- 141 00:09:08,490 --> 00:09:13,400 in physics and mechanics and engineering for springs. 142 00:09:13,400 --> 00:09:14,870 It comes up in economics. 143 00:09:14,870 --> 00:09:17,250 It's everywhere. 144 00:09:17,250 --> 00:09:21,940 And by choosing constant coefficients, I have the basic 145 00:09:21,940 --> 00:09:24,430 model and the simplest model. 146 00:09:24,430 --> 00:09:29,880 OK, so I'll talk in this language of springs, but it's 147 00:09:29,880 --> 00:09:34,420 all these oscillations that lead to equations like that. 148 00:09:34,420 --> 00:09:42,300 Actually, this model often would have r equals 0. 149 00:09:42,300 --> 00:09:46,920 So let me take that case r equals 0 again. 150 00:09:46,920 --> 00:09:50,030 So where does the equation come from? 151 00:09:50,030 --> 00:09:55,270 Can I do two cents worth of physics and then go back to 152 00:09:55,270 --> 00:09:57,880 the math, solving the equation. 153 00:09:57,880 --> 00:10:00,400 The physics is just remembering Newton's 154 00:10:00,400 --> 00:10:02,431 Law: f equals ma. 155 00:10:05,800 --> 00:10:07,970 So there is the m, the mass. 156 00:10:07,970 --> 00:10:10,690 The a is the acceleration. 157 00:10:10,690 --> 00:10:12,920 That's the second derivative, right? 158 00:10:12,920 --> 00:10:16,175 You don't mind if I write that as second derivative. 159 00:10:19,660 --> 00:10:22,390 Acceleration, the mass is constant. 160 00:10:22,390 --> 00:10:28,490 We're not going at the speed of light here so we can assume 161 00:10:28,490 --> 00:10:30,880 that mass is not being converted to energy. 162 00:10:30,880 --> 00:10:32,290 It's mass. 163 00:10:32,290 --> 00:10:34,150 And then what's the force? 164 00:10:34,150 --> 00:10:36,460 Well, for this spring force, for this 165 00:10:36,460 --> 00:10:39,180 spring, what did we say? 166 00:10:41,950 --> 00:10:44,380 So y is going this way. 167 00:10:44,380 --> 00:10:49,690 It's the disposition, the displacement of the spring. 168 00:10:49,690 --> 00:10:56,620 When it's down like that, when y is positive, the spring is 169 00:10:56,620 --> 00:10:59,200 pulling back. 170 00:10:59,200 --> 00:11:01,540 The force from the spring is pulling back. 171 00:11:01,540 --> 00:11:05,830 And the force from the spring is proportional to y. 172 00:11:05,830 --> 00:11:11,570 And the proportionality constant is my number k. 173 00:11:11,570 --> 00:11:14,360 That fact that I just said is-- 174 00:11:14,360 --> 00:11:21,400 and I guess it's pulling opposite to a positive y. 175 00:11:21,400 --> 00:11:27,450 When y is positive, and this spring is way down, the force 176 00:11:27,450 --> 00:11:28,990 is pulling it up, pulling in the 177 00:11:28,990 --> 00:11:30,970 negative direction, upwards. 178 00:11:30,970 --> 00:11:33,510 So I need that minus sign. 179 00:11:33,510 --> 00:11:38,540 So that k is the stiffness of the spring. 180 00:11:38,540 --> 00:11:42,120 This is Hooke's Law, that the force coming from a spring is 181 00:11:42,120 --> 00:11:46,260 proportional to the stretch. 182 00:11:46,260 --> 00:11:50,610 And the constant in there is Hooke's constant, the spring 183 00:11:50,610 --> 00:11:52,400 constant k. 184 00:11:52,400 --> 00:11:56,570 So now that if I put the minus ky over there as plus ky, you 185 00:11:56,570 --> 00:11:58,820 see my equation again. 186 00:11:58,820 --> 00:12:03,756 But this is the one that we were able to solve right here. 187 00:12:03,756 --> 00:12:10,070 Do you see that this is the case where r is 0 in this 188 00:12:10,070 --> 00:12:11,870 first model? 189 00:12:11,870 --> 00:12:19,150 r is 0 because r involves resistance: r for resistance, 190 00:12:19,150 --> 00:12:21,376 air resistance, damping. 191 00:12:21,376 --> 00:12:25,090 And right now, I don't have that. 192 00:12:25,090 --> 00:12:28,340 I just have a little spring that'll oscillate forever. 193 00:12:28,340 --> 00:12:34,080 It'll oscillate forever following sine and cosine. 194 00:12:34,080 --> 00:12:35,950 That's exactly what the spring will do. 195 00:12:35,950 --> 00:12:39,800 It will just go on forever, and the c and the d, their 196 00:12:39,800 --> 00:12:43,640 constants, will depend on how it started. 197 00:12:43,640 --> 00:12:46,060 Did it start from rest? 198 00:12:46,060 --> 00:12:51,630 If it started from rest, there would be no sine term. 199 00:12:51,630 --> 00:12:53,980 It's easy to find c and d later. 200 00:12:53,980 --> 00:12:57,210 The real problem is to solve the equation, and we've done 201 00:12:57,210 --> 00:12:58,820 it for this equation. 202 00:13:01,330 --> 00:13:03,940 OK, and let's just remember this. 203 00:13:03,940 --> 00:13:08,950 Let me repeat that when I put that onto the other side and 204 00:13:08,950 --> 00:13:13,400 divide by k, then I see the-- oh, divided by m, sorry-- 205 00:13:13,400 --> 00:13:15,210 then I see the omega squared. 206 00:13:15,210 --> 00:13:19,790 So omega squared is k/m. 207 00:13:19,790 --> 00:13:21,580 Right. 208 00:13:21,580 --> 00:13:24,590 Oh, it is k/m, right. 209 00:13:24,590 --> 00:13:29,130 OK, that's the simple case, pure sines and cosines. 210 00:13:29,130 --> 00:13:33,130 Now I'm letting in some resistance. 211 00:13:33,130 --> 00:13:35,370 So now I'm coming back to my equation. 212 00:13:35,370 --> 00:13:36,930 Let me write it again. 213 00:13:36,930 --> 00:13:39,770 m y double prime-- 214 00:13:39,770 --> 00:13:41,530 second derivative-- 215 00:13:41,530 --> 00:13:44,910 plus 2r y prime-- 216 00:13:44,910 --> 00:13:46,540 first derivative-- 217 00:13:46,540 --> 00:13:48,720 plus ky-- 218 00:13:48,720 --> 00:13:52,290 0-th derivative equals 0. 219 00:13:52,290 --> 00:13:55,960 I want to solve that equation now for any 220 00:13:55,960 --> 00:14:00,510 numbers m and r and k. 221 00:14:00,510 --> 00:14:08,200 OK, well, the nice thing is that the exponential function 222 00:14:08,200 --> 00:14:12,800 takes us right to the answer, the best plan here. 223 00:14:12,800 --> 00:14:16,030 So this is the most important equation you would see in a 224 00:14:16,030 --> 00:14:17,900 differential equations course. 225 00:14:17,900 --> 00:14:22,440 And then the course kind of goes past, and then you easily 226 00:14:22,440 --> 00:14:26,070 forget this is the most important and the simplest. 227 00:14:26,070 --> 00:14:27,160 Why is it simple? 228 00:14:27,160 --> 00:14:30,230 Because if the key idea-- 229 00:14:30,230 --> 00:14:36,125 this is the key idea: try y equals e to the-- 230 00:14:36,125 --> 00:14:40,140 an exponential e to the something times t. 231 00:14:40,140 --> 00:14:43,150 Let me call that something lambda. 232 00:14:43,150 --> 00:14:45,120 You might have preferred c. 233 00:14:45,120 --> 00:14:48,460 I'm happy with c or any other number there. 234 00:14:48,460 --> 00:14:52,685 Yeah, OK, I'll call that lambda, just because it gives 235 00:14:52,685 --> 00:14:54,570 it a little Greek importance. 236 00:14:54,570 --> 00:14:57,170 All right, so I try this. 237 00:14:57,170 --> 00:15:01,180 I substitute that into the equation, and I'm going to 238 00:15:01,180 --> 00:15:04,110 choose lambda to make things work. 239 00:15:04,110 --> 00:15:07,160 OK, but here's the key. 240 00:15:07,160 --> 00:15:09,000 The key idea is so easy. 241 00:15:09,000 --> 00:15:10,760 Now, put it into the equation. 242 00:15:10,760 --> 00:15:13,220 So what happens when I put-- 243 00:15:13,220 --> 00:15:16,100 when this is y, take two derivatives. 244 00:15:16,100 --> 00:15:18,800 Well, let me start with taking no derivatives. 245 00:15:18,800 --> 00:15:23,500 So I have down here the k e to the lambda t, and 246 00:15:23,500 --> 00:15:25,170 over here is a 0. 247 00:15:25,170 --> 00:15:29,160 And now let me back up to the first derivative. 248 00:15:29,160 --> 00:15:33,650 So that's 2r times the derivative 249 00:15:33,650 --> 00:15:35,460 of this guy y prime. 250 00:15:35,460 --> 00:15:38,940 I'm just substituting this into the equation. 251 00:15:38,940 --> 00:15:41,480 So what's the derivative? 252 00:15:41,480 --> 00:15:45,360 We know that the derivative of this brings down the lambda. 253 00:15:45,360 --> 00:15:46,680 We already did it. 254 00:15:46,680 --> 00:15:50,130 lambda e to the lambda t, right? 255 00:15:50,130 --> 00:15:51,770 That's the derivative. 256 00:15:51,770 --> 00:15:53,550 And what about this one? 257 00:15:53,550 --> 00:15:57,980 This one is going to be an m y double prime. 258 00:15:57,980 --> 00:16:00,520 What happens with two derivatives? 259 00:16:00,520 --> 00:16:05,160 Bring down lambda twice, two times, so I have lambda 260 00:16:05,160 --> 00:16:09,140 squared e to the lambda t. 261 00:16:09,140 --> 00:16:14,430 And then in a minute, you know what I'm going to do. 262 00:16:14,430 --> 00:16:18,900 I'm going to cancel that common factor e to the lambda 263 00:16:18,900 --> 00:16:24,140 t, which is never 0 so I can safely divide it out, and then 264 00:16:24,140 --> 00:16:26,690 write the equation we get. 265 00:16:26,690 --> 00:16:29,470 OK, let me write that out more with more space. 266 00:16:29,470 --> 00:16:36,260 m lambda squared plus 2r lambda-- 267 00:16:36,260 --> 00:16:37,230 taking that-- 268 00:16:37,230 --> 00:16:41,435 plus k is 0. 269 00:16:44,140 --> 00:16:48,120 This is the equation. 270 00:16:48,120 --> 00:16:51,640 It's just an ordinary quadratic equation. 271 00:16:51,640 --> 00:16:55,730 It's a high school algebra equation. 272 00:16:55,730 --> 00:16:59,740 Lambda appears squared because we had two derivatives. 273 00:16:59,740 --> 00:17:03,920 We had a second derivative, and so I need the quadratic 274 00:17:03,920 --> 00:17:06,130 formula to know-- 275 00:17:06,130 --> 00:17:09,410 I expect two answers, two lambdas. 276 00:17:09,410 --> 00:17:14,220 And that's normal for a second order equation, and I'll get 277 00:17:14,220 --> 00:17:19,329 two solutions: e to the lambda 1t and e to the lambda 2t. 278 00:17:19,329 --> 00:17:23,750 Two different exponentials will both solve the problem. 279 00:17:23,750 --> 00:17:27,560 All right, what's the lambda? 280 00:17:27,560 --> 00:17:31,380 Well, can I just recall the quadratic formula? 281 00:17:31,380 --> 00:17:35,160 Well, it's a little messy, but it's not too bad here, just to 282 00:17:35,160 --> 00:17:39,280 show that I remember it. 283 00:17:39,280 --> 00:17:42,770 And the 2 there is kind of handy with the quadratic 284 00:17:42,770 --> 00:17:47,730 formula because then I just get minus an r plus or minus 285 00:17:47,730 --> 00:17:51,370 the square root of r squared. 286 00:17:51,370 --> 00:17:55,830 And it's not minus 4km, but because of the 2 there, it's 287 00:17:55,830 --> 00:17:57,080 just minus km. 288 00:17:59,440 --> 00:18:04,680 And then I divide by m. 289 00:18:04,680 --> 00:18:10,280 OK, well, big deal. 290 00:18:10,280 --> 00:18:12,710 I get two roots. 291 00:18:12,710 --> 00:18:14,280 Let me use numbers. 292 00:18:14,280 --> 00:18:19,020 So what you see there is like solving the 293 00:18:19,020 --> 00:18:20,860 differential equation. 294 00:18:20,860 --> 00:18:22,410 Not too bad. 295 00:18:22,410 --> 00:18:27,990 Now let me put in numbers to show what's typical, and, of 296 00:18:27,990 --> 00:18:31,200 course, as those numbers change, we'll 297 00:18:31,200 --> 00:18:35,710 see different lambdas. 298 00:18:35,710 --> 00:18:38,820 And actually, as the numbers change, that will take us 299 00:18:38,820 --> 00:18:44,330 between the exponential stuff and the oscillating stuff. 300 00:18:44,330 --> 00:18:48,270 All right, let me take one where I think it start-- 301 00:18:48,270 --> 00:18:50,770 I think this will be-- so this is example one. 302 00:18:54,310 --> 00:19:01,640 I'll choose m equal 1 y double prime plus-- 303 00:19:01,640 --> 00:19:07,110 let me take r to be 3, so then I have 6y prime. 304 00:19:07,110 --> 00:19:13,260 And let me choose k to be 8y equals 0. 305 00:19:13,260 --> 00:19:15,370 All right, now we've got numbers. 306 00:19:19,050 --> 00:19:25,690 So the numbers are m equals 1, r equals 3, k equals 8. 307 00:19:25,690 --> 00:19:30,740 And I claim we can write the solution to that equation, or 308 00:19:30,740 --> 00:19:33,850 the two solutions, because there will be two lambdas. 309 00:19:33,850 --> 00:19:40,400 OK, so when I try e to the lambda t, plug it in, I'll get 310 00:19:40,400 --> 00:19:46,220 lambda twice, and then I'll get 6 lambda once, and then 311 00:19:46,220 --> 00:19:50,510 I'll get 8 without a lambda coming down, all multiplied by 312 00:19:50,510 --> 00:19:53,540 e to the lambda t, which I'm canceling, equals 0. 313 00:19:56,090 --> 00:20:03,000 So I solve that equation either directly by recognizing 314 00:20:03,000 --> 00:20:07,960 that it factors into lambda plus 2 times lambda plus 4 315 00:20:07,960 --> 00:20:13,310 equals 0 or by plugging in r and k and m in 316 00:20:13,310 --> 00:20:14,640 the quadratic formula. 317 00:20:14,640 --> 00:20:22,450 Either way, I'm learning that lambda is minus 2 or minus 4. 318 00:20:22,450 --> 00:20:24,720 Those are the two solutions. 319 00:20:24,720 --> 00:20:29,330 The two decay rates, you could call them, because they're up 320 00:20:29,330 --> 00:20:30,820 in the exponent. 321 00:20:30,820 --> 00:20:32,140 So what's the solution? 322 00:20:32,140 --> 00:20:39,670 y of t, the solution to that equation, the general solution 323 00:20:39,670 --> 00:20:45,370 with a constant c and a constant d is an e to the 324 00:20:45,370 --> 00:20:52,680 minus 2t and an e to the minus 4t. 325 00:20:52,680 --> 00:20:58,960 The two lambdas are in the exponent, and we've solved it. 326 00:20:58,960 --> 00:21:01,190 So that's the point. 327 00:21:01,190 --> 00:21:07,380 We have the ability to solve differential equations based 328 00:21:07,380 --> 00:21:12,560 on the three most important derivatives we know: 329 00:21:12,560 --> 00:21:16,590 exponential, sines-cosines, powers of t. 330 00:21:16,590 --> 00:21:19,070 OK, ready for example two? 331 00:21:19,070 --> 00:21:23,370 Example two, I'm just going to change that 8 to a 10, so 332 00:21:23,370 --> 00:21:26,280 you're going to see a 10 show up here. 333 00:21:26,280 --> 00:21:27,915 All right, but that will make a difference. 334 00:21:27,915 --> 00:21:30,460 It won't just be some new numbers. 335 00:21:30,460 --> 00:21:33,410 There'll be a definite difference here. 336 00:21:33,410 --> 00:21:38,630 OK, let me go over across here to the one with 10. 337 00:21:38,630 --> 00:21:46,210 OK, so now my equation is 1y double prime, 6y prime still, 338 00:21:46,210 --> 00:21:51,660 and now 10y is equals to 0, remembering prime means 339 00:21:51,660 --> 00:21:53,220 derivative. 340 00:21:53,220 --> 00:21:58,470 OK, so again, I try y is e to the lambda t. 341 00:21:58,470 --> 00:21:59,940 I plug it in. 342 00:21:59,940 --> 00:22:03,290 When I have two derivatives, bring down lambda squared. 343 00:22:03,290 --> 00:22:06,140 One derivative brings down lambda. 344 00:22:06,140 --> 00:22:09,930 No derivatives leaves the 10, equals 0. 345 00:22:09,930 --> 00:22:13,860 That's my equation for lambda. 346 00:22:13,860 --> 00:22:14,870 Ha! 347 00:22:14,870 --> 00:22:17,730 I don't know how to factor that one. 348 00:22:17,730 --> 00:22:24,930 And in fact, I better use the quadratic formula just to show 349 00:22:24,930 --> 00:22:26,420 what happens here. 350 00:22:26,420 --> 00:22:31,350 So the quadratic formula will be the two roots lambda. 351 00:22:35,580 --> 00:22:38,500 Can I remember that dumb formula? 352 00:22:38,500 --> 00:22:43,290 Minus r plus or minus the square root of r squared minus 353 00:22:43,290 --> 00:22:47,010 km, all divided by m. 354 00:22:47,010 --> 00:22:51,690 I got the 2's and the 4's out of it by 355 00:22:51,690 --> 00:22:54,080 taking r to be 3 here. 356 00:22:54,080 --> 00:23:01,850 OK, so it's minus 3 plus or minus the square root of r 357 00:23:01,850 --> 00:23:07,820 squared is 9 minus 1 times 10. 358 00:23:07,820 --> 00:23:13,394 k and m is 10 divided by 1. 359 00:23:13,394 --> 00:23:15,270 Ha! 360 00:23:15,270 --> 00:23:18,570 You see something different's going on here. 361 00:23:18,570 --> 00:23:22,240 I have the square root of a negative number. 362 00:23:22,240 --> 00:23:27,430 Over there, if I wrote out that square root, you would 363 00:23:27,430 --> 00:23:30,910 have seen the square root of plus 1. 364 00:23:30,910 --> 00:23:34,900 That gave me minus 3 plus 1 or minus 3 minus 1. 365 00:23:34,900 --> 00:23:37,870 That was the minus 2 and minus 4. 366 00:23:37,870 --> 00:23:39,870 Now I'm different. 367 00:23:39,870 --> 00:23:45,020 Now seeing the square root of minus 1, so this is minus 3 368 00:23:45,020 --> 00:23:46,920 plus or minus i. 369 00:23:46,920 --> 00:23:52,460 So I see the solution y of t. 370 00:23:52,460 --> 00:23:57,645 You see, this is i here, the square root of minus 1. 371 00:24:02,110 --> 00:24:04,610 We can deal with that. 372 00:24:04,610 --> 00:24:07,800 It's a complex number, an imaginary number. 373 00:24:07,800 --> 00:24:11,600 And the combination minus 3 plus i is a complex number, 374 00:24:11,600 --> 00:24:14,690 and we have to accept that that's our lambda. 375 00:24:14,690 --> 00:24:20,760 So I have any multiple of c, and the lambda here is 376 00:24:20,760 --> 00:24:26,060 minus 3 plus i t. 377 00:24:26,060 --> 00:24:33,980 And the second solution is e minus 3 minus i t. 378 00:24:38,290 --> 00:24:39,560 I found the general solution. 379 00:24:42,880 --> 00:24:48,770 We could say done, except you might feel, well, how did 380 00:24:48,770 --> 00:24:49,660 imaginary-- 381 00:24:49,660 --> 00:24:53,040 what are we going to do with these imaginary numbers here? 382 00:24:53,040 --> 00:24:55,460 How did they get in this perfectly real 383 00:24:55,460 --> 00:24:57,210 differential equation? 384 00:24:57,210 --> 00:25:01,360 Well, they slipped in because the solutions 385 00:25:01,360 --> 00:25:03,110 were not real numbers. 386 00:25:03,110 --> 00:25:06,560 The solutions were minus 3 plus or minus i. 387 00:25:06,560 --> 00:25:10,290 But we can get real again. 388 00:25:10,290 --> 00:25:14,330 So this is one way to write the answer, but I just want to 389 00:25:14,330 --> 00:25:20,870 show you using the earlier lecture, using the beautiful 390 00:25:20,870 --> 00:25:24,660 fact that Euler discovered. 391 00:25:24,660 --> 00:25:29,350 So now let me complete this example by remembering Euler's 392 00:25:29,350 --> 00:25:32,870 great formula for e to the it. 393 00:25:32,870 --> 00:25:37,790 Because you see we have an e to the minus 3t, perfectly 394 00:25:37,790 --> 00:25:39,650 real, decaying. 395 00:25:39,650 --> 00:25:43,950 The spring is slowing down because of air resistance. 396 00:25:43,950 --> 00:25:47,910 But we also have an e to the it. 397 00:25:47,910 --> 00:25:51,840 That's what Euler's formulas about and Euler's formula says 398 00:25:51,840 --> 00:25:57,550 that e to the it is the cosine of t plus i 399 00:25:57,550 --> 00:26:01,220 times the sine of t. 400 00:26:01,220 --> 00:26:04,620 So it's through Euler's formula that these 401 00:26:04,620 --> 00:26:06,500 oscillations are coming in. 402 00:26:06,500 --> 00:26:10,800 The direct method led to an e to the it. 403 00:26:10,800 --> 00:26:18,640 But the next day, Euler realized that e 404 00:26:18,640 --> 00:26:20,420 to the minus it-- 405 00:26:20,420 --> 00:26:24,970 or probably being Euler, it didn't take till next day-- 406 00:26:24,970 --> 00:26:29,580 will be minus i sine t. 407 00:26:29,580 --> 00:26:36,080 So both e to the it and e to the minus it, they both can 408 00:26:36,080 --> 00:26:39,780 get replaced by sines and cosines. 409 00:26:39,780 --> 00:26:43,460 So in place of e to the it, I'll put that. 410 00:26:43,460 --> 00:26:46,080 In place of e to the minus it, I put that one. 411 00:26:46,080 --> 00:26:48,160 The final result is-- 412 00:26:48,160 --> 00:26:50,200 can I just jump to that? 413 00:26:50,200 --> 00:26:54,730 The final result is that with some different constants, we 414 00:26:54,730 --> 00:26:56,080 have the cosine-- 415 00:26:56,080 --> 00:26:57,140 oh! 416 00:26:57,140 --> 00:27:00,700 let me not forget e to the minus 3t. 417 00:27:00,700 --> 00:27:03,510 That's part of this answer. 418 00:27:03,510 --> 00:27:08,010 I'm damping this out by e to the minus 3t, this resistance 419 00:27:08,010 --> 00:27:18,990 r, times cosine of t and the e to the minus 3t sine of t. 420 00:27:18,990 --> 00:27:21,055 OK, that's good. 421 00:27:23,630 --> 00:27:29,080 General solution, back to a real numbers. 422 00:27:29,080 --> 00:27:32,310 It describes a damped oscillation. 423 00:27:32,310 --> 00:27:33,650 It's damped out. 424 00:27:33,650 --> 00:27:35,290 It's slowing down. 425 00:27:35,290 --> 00:27:37,780 Rather, it's decaying. 426 00:27:37,780 --> 00:27:40,940 The amplitude is-- the spring is like-- 427 00:27:40,940 --> 00:27:43,250 it's like having a shock absorber or something. 428 00:27:43,250 --> 00:27:50,760 It's settling down to the center point pretty fast. But 429 00:27:50,760 --> 00:27:54,170 as it settles, it's goes back across that center point, 430 00:27:54,170 --> 00:27:55,680 oscillates across. 431 00:27:55,680 --> 00:28:01,250 OK, now you might finally ask, the last step of this lecture, 432 00:28:01,250 --> 00:28:06,010 where do powers of t come in? 433 00:28:06,010 --> 00:28:07,580 Where does t come in? 434 00:28:07,580 --> 00:28:11,470 So far we've seen exponentials come in. 435 00:28:11,470 --> 00:28:14,180 We've seen sines and cosines come in. 436 00:28:14,180 --> 00:28:19,090 Can I do a last example just here in the corner, which will 437 00:28:19,090 --> 00:28:25,980 be y double prime, 6 y prime, and now this time instead of 8 438 00:28:25,980 --> 00:28:31,850 or 10, I'm going to take 9y equals 0. 439 00:28:31,850 --> 00:28:38,610 OK, can we use this as example 3, which we can solve? 440 00:28:38,610 --> 00:28:42,880 You know that I'm going to try y equals to lambda t. 441 00:28:42,880 --> 00:28:45,360 Let me substitute that. 442 00:28:45,360 --> 00:28:49,790 I'll get lambda squared coming from two derivatives, lambda 443 00:28:49,790 --> 00:28:55,930 coming from one derivative, 9 coming from no derivatives. 444 00:28:55,930 --> 00:28:59,400 I've got my quadratic equation that's supposed 445 00:28:59,400 --> 00:29:02,210 to give me two lambdas. 446 00:29:02,210 --> 00:29:03,790 Little problem here. 447 00:29:03,790 --> 00:29:08,100 When I factor this, it factors into lambda plus 3 448 00:29:08,100 --> 00:29:10,900 squared equals 0. 449 00:29:10,900 --> 00:29:16,440 So the answer, the lambda, is minus 3 twice. 450 00:29:19,620 --> 00:29:21,400 Twice! 451 00:29:21,400 --> 00:29:28,810 The two lambdas happen to hit the same value: minus 3. 452 00:29:28,810 --> 00:29:32,730 OK, we don't have any complex stuff here. 453 00:29:32,730 --> 00:29:36,790 It's two real values that happen to coincide. 454 00:29:36,790 --> 00:29:42,120 And when that happens, well, minus 3 tells us that a 455 00:29:42,120 --> 00:29:47,240 solution e to the minus 3t works. 456 00:29:47,240 --> 00:29:50,360 Where do we get the other solution? 457 00:29:50,360 --> 00:29:54,260 We want two solutions to a second order equation. 458 00:29:54,260 --> 00:29:58,040 I can't just use e to the minus 3t again. 459 00:29:58,040 --> 00:30:02,580 I need a second solution, and it shows up at this-- 460 00:30:02,580 --> 00:30:05,560 it's typical of math that it shows-- 461 00:30:05,560 --> 00:30:08,960 something special happens at this special situation of a 462 00:30:08,960 --> 00:30:16,440 double root, and the solutions will be e to the minus 3t and 463 00:30:16,440 --> 00:30:19,170 t e to the minus 3t. 464 00:30:19,170 --> 00:30:22,490 That's the last step, and maybe I just put it-- 465 00:30:22,490 --> 00:30:25,670 well, where am I going to put it? 466 00:30:25,670 --> 00:30:29,120 I'll bring this down and just put it here. 467 00:30:29,120 --> 00:30:34,030 So now I'm solving this particular problem y is a 468 00:30:34,030 --> 00:30:37,330 multiple of e to the minus 3t. 469 00:30:37,330 --> 00:30:42,340 Good, but I can't just repeat it for the second one. 470 00:30:42,340 --> 00:30:46,160 So the second one, it just turns out that then is when a 471 00:30:46,160 --> 00:30:47,490 little factor t appears. 472 00:30:50,030 --> 00:30:54,760 You might like to substitute that for practice with the 473 00:30:54,760 --> 00:30:56,220 product rule. 474 00:30:56,220 --> 00:30:59,280 If you substitute that in the differential equation, you'll 475 00:30:59,280 --> 00:31:04,360 find everything cancels, and it works. 476 00:31:04,360 --> 00:31:11,480 So the conclusion is linear constant coefficient 477 00:31:11,480 --> 00:31:17,030 differential equations are completely solved by trying e 478 00:31:17,030 --> 00:31:22,140 to the lambda t and finding that number lambda. 479 00:31:22,140 --> 00:31:25,950 If it's a real number, we have exponentials. 480 00:31:25,950 --> 00:31:29,810 If it's an imaginary number, we have sines and cosines. 481 00:31:29,810 --> 00:31:33,950 If it's a repeated number, we have an extra 482 00:31:33,950 --> 00:31:36,650 factor t showing up. 483 00:31:36,650 --> 00:31:42,690 That's the exceptional event in that particular case. 484 00:31:42,690 --> 00:31:43,890 So there you go. 485 00:31:43,890 --> 00:31:46,950 Differential equations with constant 486 00:31:46,950 --> 00:31:50,140 coefficients we can handle. 487 00:31:50,140 --> 00:31:52,040 You can handle. 488 00:31:52,040 --> 00:31:53,290 Thank you. 489 00:31:55,090 --> 00:31:56,900 This has been a production of MIT 490 00:31:56,900 --> 00:31:59,290 OpenCourseWare and Gilbert Strang. 491 00:31:59,290 --> 00:32:01,560 Funding for this video was provided by the Lord 492 00:32:01,560 --> 00:32:02,780 Foundation. 493 00:32:02,780 --> 00:32:05,910 To help OCW continue to provide free and open access 494 00:32:05,910 --> 00:32:08,990 to MIT courses, please make a donation at 495 00:32:08,990 --> 00:32:10,550 ocw.mit.edu/donate.